I'm having difficulties proving that End(V) is a k-algebra. First, I need to prove that End(V) i.e. Hom(V,V) is a k- vector space but under what operation? I think I have to prove that End(V) is a k-algebra under the composition operation but do I also have to prove that Hom(V,V) is a vector space under the composition operation?

For example: a part of the definition a vector space in my Professor's lecture notes is:

1) a(X+Y) = aX + aY for all a in K and X,Y in V.so on and so forth. But End(V) is perhaps not a k-algebra under Addition and Multiplication...so I have to proof that the Hom(V,V) is a vector space under the composition operation?

Maybe I'm over thinking it...but nonetheless it would be great if I would work with someone through this problem.